Would that imply that s_k has no limit as well? Or is that not enough?

Actually, you've provided "too much"! See, the statement "If [tex]s_{2k}[/tex] has no limit, then neither does [tex]s_k[/tex]" is the contrapositive of "If [tex]s_k[/tex] has a limit, then so does [tex]s_{2k}[/tex]," the truth of which follows quite readily for all sequences from the definition of a sequence limit.